PEOFESSOE CAYLEY ON PBEPOTENTIALS. 
731 
89. For the reduction of the first and fourth of these we have to consider 
-(l-#)n(«, 0, 1-0, v)+~ n(«-l, 0+1, -0, v); 
viz. this is 
( — 1 +v + l+v)^ x . 1 -vxY~ l dx— 2 ^ vx . x a ~\ 1—x. 1 —vxf-'dx, 
=2v . . 1 -vxf-'dx, 
=2 v . II(a, 3 + 1, — 3+1, v ) ; 
that is, 
-(i-»)n(«,0,i-0,»)+^n( a -i,0+i, -0, !) )=2®n(«,0+i, -0+1, »). 
[I repeat for comparison the foregoing equation, 
+(1 -V) n(a, 0, 1-0, «)+?=i n(a-l, 0+1, -0, «)=2II(a, 0, -0, V); 
by adding and subtracting these we obtain two new formulae ] ; for reduction of the 
fourth formula the equation may be written 
-n(«-i, 0+1, -0, v)+(i-v)£i . n(«, 0, 1-0, ®)=-2 At vn(a, 0 + i-0+i, 4 
90. But it is sufficient to consider the first formula ; the term in [ ] is 
=r|^nh> (^)' +!, ' a 5 n(*+* *-**+*$, 
and the corresponding value of the disk-integral is 
rjsrj + +I 
which we may again verify by taking therein z indefinitely large ; viz. the value is then 
pi s pj, fs+l 
= f(fr+|) «*+ as a bove. It is the first of a system of four forms, the others of which 
are 
r+r| + +1 
r (i+?)r(+-?+i) * s+2? 
,/ s+i 
■■ r(|s+?)r(|-g') x*+ 2 ? 
y^+i 
~r(is-g + l)r(^ + g') 
i s +2,~2^5 
o-r) ’ 
And hence, writing as before x= 
t+P-K? 
t . 
&c., the four values are 
5 E 
MDCCCLXXV. 
